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Register a new userA family of Exponential Integrals suggested by Stellar Dynamics
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While investigating the generalization of the Chandrasekhar (1943) dynamical friction to the case of field stars with a power-law mass spectrum and equipartition Maxwell-Boltzmann velocity distribution, a pair of 2-dimensional integrals involving the Error function occurred, with closed form solution in terms of Exponential Integrals (Ciotti 2010). Here we show that both the integrals are very special cases of the family of (real) functions $$ I(lambda,mu, u; z) :=int_0^zx^{lambda},Enu(x^{mu}),dx= {gammaleft({1+lambdaovermu},z^{mu}right) + z^{1+lambda}Enu(z^{mu})over 1+lambda + mu ( u -1)}, quad mu>0,quad zgeq 0, eqno (1) $$ where $Enu$ is the Exponential Integral, $gamma$ is the incomplete Euler gamma function, and for existence $lambda >max left{-1,-1- mu( u -1)right}$. Only in one of the consulted tables a related integral appears, that with some work can be reduced to eq.~(1), while computer algebra systems seem to be able to evaluate the integral in closed (and more complicated) form only provided numerical values for some of the parameters are assigned. Here we show how eq.~(1) can in fact be established by elementary methods.
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While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$ I_n(z)equivint_0^1{{rm K}(k) kover (z+k^2)^{n+3/2}}dk = {(-2)^nover (2n+1)!!} {d^nover dz^n} {{rm ArcCot}sqrt{z}oversqrt{z(z+1)}},quad z>0,$$ where $n=0,1,2,3...$, and ${rm K}(k)$ is the complete elliptic integral of first kind.
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